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PrefaceAcronymsChapter 1 Elementary Differential Geometry1．1 Parametric Representation of Surfaces1．2 Curvatures of Surfaces1．3 The Fundamental Equations and the Fundamental Theorem of Surfaces1．4 Gauss-Bonnet Theorem1．5 Differential Operators on Surfaces1．6 Basic Properties of Differential Operators1．7 Differential Operators Acting on Surface and Normal Vector1．8 Some Global Properties of Surfaces1．8．1 Green's Formulas1．8．2 Integral Formulas of Surfaces1．9 Differential Geometry of Implicit SurfacesChapter 2 Construction of Geometric Partial Differential Equations for Parametric Surfaces2．1 Variation of Functionals for Parametric Surfaces2．2 The Second-order Euler-Lagrange Operator2．3 The Fourth-order Euler-Lagrange Operator2．4 The Sixth-order Euler-Lagrange Operator． '2．5 Other Euler-Lagrange Operators2．5．1 Additivity ofEuler-Lagrange Operators2．5．2 Euler-Lagrange Operator for Surfaces with Graph Representation2．6 GradientFlow2．6．1 L2-Gradient Flow for Parametric Surfaces2．6．2 H-1-Gradient Flow for Parametric Surfaces2．7 Other Geometric Flows2．7．1 Area-Preserving or Volume-Preserving Second-order Geometric Flows2．7．2 Other Sixth-order Geometric Flows2．7．3 Geometric Flow for Surfaces with Graph Representation2．8 Notes2．9 Related Works2．9．1 The Choice of Energy Functionals2．9．2 About Geometric FlowsChapter 3 Construction of Geometric Partial Differential Equations for Level-Set Surfaces3．1 Variation of Functionals on Level-Set Surfaces3．2 The Second-order Euler-Lagrange Operator3．3 The Fourth-order Euler-Lagrange Operator3．4 The Sixth-order Euler-Lagrange Operator3．5 L2-Gradient Flows for Level Sets3．6 H-1-Gradient Flow for Level Sets3．7 Construction of Geometric Flows from Operator Conversion3．8 Relationship Among Three Construction Methods of the Geometric FlowsChapter 4 Discretization of Differential Geometric Operators and Curvatures4．1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes4．1．1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes4．1．2 Convergence Test of Different Discretization Schemes of the LB Operator4．1．3 Convergence of the Discrete LB Operator over Triangular Meshes4．1．4 Proof of the Convergence Results4．2 Discretization of the Laplace-Beltrami Operator over Quadrilateral Meshes and Its Convergence Analysis4．2．1 Discretization of LB Operator over Quadrilateral Meshes4．2．2 Convergence Property of the Discrete LB Operator4．2．3 Simplified Integration Rule4．2．4 Numerical Experiments4．3 Discretization of the Gaussian Curvature over Triangular Meshes4．3．1 Discretization of the Gaussian Curvature over Triangular Meshes4．3．2 Numerical Experiments4．3．3 Convergence Properties of the Discrete Gaussian Curvatures4．3．4 Modified Gauss-Bonnet Schemes and Their Convergence4．3．5 A Counterexample for the Regular Vertex with Valence 44．4 Discretization of the Gaussian Curvature over Quadrilateral Meshes andIts Convergence Analysis4．4．1 Discretization of the Gaussian Curvature over Quadrilateral Meshes4．4．2 Convergence Property of the Discrete Gaussian Curvature4．5 Consistent Approximations of Some Geometric Differential Operators4．5．1 Consistent Discretizations of Differential Geometric Operators and Curvatures Based on the Quadratic Fitting of Surfaces4．5．2 Convergence Property of Discrete Differential Operators4．5．3 Consistent Discretization of Differential Operators Based on BiquadraticInterpolation…… Chapter 5 Discrete Surface Design by Quasi Finite Difference MethodChapter 6 Spline Surface Design by Quasi Finite Difference Method and FiniteElement MethodChapter 7 Subdivision Surface Dqesign by Finite Element MethodsChapter 8 Level-Set Method for Surface Design and Its ApplicationsChapter 9 Quality Meshing with Geometric FlowsReferencesIndex